Optimal. Leaf size=276 \[ \frac {\sin \left (a-\frac {b \sqrt {f}}{\sqrt {c f-d e}}\right ) \text {Ci}\left (\frac {\sqrt {f} b}{\sqrt {c f-d e}}+\frac {b}{\sqrt {c+d x}}\right )}{f}+\frac {\sin \left (a+\frac {b \sqrt {f}}{\sqrt {c f-d e}}\right ) \text {Ci}\left (\frac {b \sqrt {f}}{\sqrt {c f-d e}}-\frac {b}{\sqrt {c+d x}}\right )}{f}-\frac {2 \sin (a) \text {Ci}\left (\frac {b}{\sqrt {c+d x}}\right )}{f}-\frac {\cos \left (a+\frac {b \sqrt {f}}{\sqrt {c f-d e}}\right ) \text {Si}\left (\frac {b \sqrt {f}}{\sqrt {c f-d e}}-\frac {b}{\sqrt {c+d x}}\right )}{f}+\frac {\cos \left (a-\frac {b \sqrt {f}}{\sqrt {c f-d e}}\right ) \text {Si}\left (\frac {\sqrt {f} b}{\sqrt {c f-d e}}+\frac {b}{\sqrt {c+d x}}\right )}{f}-\frac {2 \cos (a) \text {Si}\left (\frac {b}{\sqrt {c+d x}}\right )}{f} \]
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Rubi [A] time = 1.20, antiderivative size = 276, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {3431, 3303, 3299, 3302, 3345} \[ \frac {\sin \left (a-\frac {b \sqrt {f}}{\sqrt {c f-d e}}\right ) \text {CosIntegral}\left (\frac {b \sqrt {f}}{\sqrt {c f-d e}}+\frac {b}{\sqrt {c+d x}}\right )}{f}+\frac {\sin \left (a+\frac {b \sqrt {f}}{\sqrt {c f-d e}}\right ) \text {CosIntegral}\left (\frac {b \sqrt {f}}{\sqrt {c f-d e}}-\frac {b}{\sqrt {c+d x}}\right )}{f}-\frac {2 \sin (a) \text {CosIntegral}\left (\frac {b}{\sqrt {c+d x}}\right )}{f}-\frac {\cos \left (a+\frac {b \sqrt {f}}{\sqrt {c f-d e}}\right ) \text {Si}\left (\frac {b \sqrt {f}}{\sqrt {c f-d e}}-\frac {b}{\sqrt {c+d x}}\right )}{f}+\frac {\cos \left (a-\frac {b \sqrt {f}}{\sqrt {c f-d e}}\right ) \text {Si}\left (\frac {\sqrt {f} b}{\sqrt {c f-d e}}+\frac {b}{\sqrt {c+d x}}\right )}{f}-\frac {2 \cos (a) \text {Si}\left (\frac {b}{\sqrt {c+d x}}\right )}{f} \]
Antiderivative was successfully verified.
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Rule 3299
Rule 3302
Rule 3303
Rule 3345
Rule 3431
Rubi steps
\begin {align*} \int \frac {\sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{e+f x} \, dx &=-\frac {2 \operatorname {Subst}\left (\int \left (\frac {d \sin (a+b x)}{f x}+\frac {d (-d e+c f) x \sin (a+b x)}{f \left (f+(d e-c f) x^2\right )}\right ) \, dx,x,\frac {1}{\sqrt {c+d x}}\right )}{d}\\ &=-\frac {2 \operatorname {Subst}\left (\int \frac {\sin (a+b x)}{x} \, dx,x,\frac {1}{\sqrt {c+d x}}\right )}{f}+\frac {(2 (d e-c f)) \operatorname {Subst}\left (\int \frac {x \sin (a+b x)}{f+(d e-c f) x^2} \, dx,x,\frac {1}{\sqrt {c+d x}}\right )}{f}\\ &=\frac {(2 (d e-c f)) \operatorname {Subst}\left (\int \left (-\frac {\sqrt {-d e+c f} \sin (a+b x)}{2 (d e-c f) \left (\sqrt {f}-\sqrt {-d e+c f} x\right )}+\frac {\sqrt {-d e+c f} \sin (a+b x)}{2 (d e-c f) \left (\sqrt {f}+\sqrt {-d e+c f} x\right )}\right ) \, dx,x,\frac {1}{\sqrt {c+d x}}\right )}{f}-\frac {(2 \cos (a)) \operatorname {Subst}\left (\int \frac {\sin (b x)}{x} \, dx,x,\frac {1}{\sqrt {c+d x}}\right )}{f}-\frac {(2 \sin (a)) \operatorname {Subst}\left (\int \frac {\cos (b x)}{x} \, dx,x,\frac {1}{\sqrt {c+d x}}\right )}{f}\\ &=-\frac {2 \text {Ci}\left (\frac {b}{\sqrt {c+d x}}\right ) \sin (a)}{f}-\frac {2 \cos (a) \text {Si}\left (\frac {b}{\sqrt {c+d x}}\right )}{f}-\frac {\sqrt {-d e+c f} \operatorname {Subst}\left (\int \frac {\sin (a+b x)}{\sqrt {f}-\sqrt {-d e+c f} x} \, dx,x,\frac {1}{\sqrt {c+d x}}\right )}{f}+\frac {\sqrt {-d e+c f} \operatorname {Subst}\left (\int \frac {\sin (a+b x)}{\sqrt {f}+\sqrt {-d e+c f} x} \, dx,x,\frac {1}{\sqrt {c+d x}}\right )}{f}\\ &=-\frac {2 \text {Ci}\left (\frac {b}{\sqrt {c+d x}}\right ) \sin (a)}{f}-\frac {2 \cos (a) \text {Si}\left (\frac {b}{\sqrt {c+d x}}\right )}{f}+\frac {\left (\sqrt {-d e+c f} \cos \left (a-\frac {b \sqrt {f}}{\sqrt {-d e+c f}}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {b \sqrt {f}}{\sqrt {-d e+c f}}+b x\right )}{\sqrt {f}+\sqrt {-d e+c f} x} \, dx,x,\frac {1}{\sqrt {c+d x}}\right )}{f}+\frac {\left (\sqrt {-d e+c f} \cos \left (a+\frac {b \sqrt {f}}{\sqrt {-d e+c f}}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {b \sqrt {f}}{\sqrt {-d e+c f}}-b x\right )}{\sqrt {f}-\sqrt {-d e+c f} x} \, dx,x,\frac {1}{\sqrt {c+d x}}\right )}{f}+\frac {\left (\sqrt {-d e+c f} \sin \left (a-\frac {b \sqrt {f}}{\sqrt {-d e+c f}}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {b \sqrt {f}}{\sqrt {-d e+c f}}+b x\right )}{\sqrt {f}+\sqrt {-d e+c f} x} \, dx,x,\frac {1}{\sqrt {c+d x}}\right )}{f}-\frac {\left (\sqrt {-d e+c f} \sin \left (a+\frac {b \sqrt {f}}{\sqrt {-d e+c f}}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {b \sqrt {f}}{\sqrt {-d e+c f}}-b x\right )}{\sqrt {f}-\sqrt {-d e+c f} x} \, dx,x,\frac {1}{\sqrt {c+d x}}\right )}{f}\\ &=-\frac {2 \text {Ci}\left (\frac {b}{\sqrt {c+d x}}\right ) \sin (a)}{f}+\frac {\text {Ci}\left (\frac {b \sqrt {f}}{\sqrt {-d e+c f}}+\frac {b}{\sqrt {c+d x}}\right ) \sin \left (a-\frac {b \sqrt {f}}{\sqrt {-d e+c f}}\right )}{f}+\frac {\text {Ci}\left (\frac {b \sqrt {f}}{\sqrt {-d e+c f}}-\frac {b}{\sqrt {c+d x}}\right ) \sin \left (a+\frac {b \sqrt {f}}{\sqrt {-d e+c f}}\right )}{f}-\frac {2 \cos (a) \text {Si}\left (\frac {b}{\sqrt {c+d x}}\right )}{f}-\frac {\cos \left (a+\frac {b \sqrt {f}}{\sqrt {-d e+c f}}\right ) \text {Si}\left (\frac {b \sqrt {f}}{\sqrt {-d e+c f}}-\frac {b}{\sqrt {c+d x}}\right )}{f}+\frac {\cos \left (a-\frac {b \sqrt {f}}{\sqrt {-d e+c f}}\right ) \text {Si}\left (\frac {b \sqrt {f}}{\sqrt {-d e+c f}}+\frac {b}{\sqrt {c+d x}}\right )}{f}\\ \end {align*}
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Mathematica [F] time = 15.69, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{e+f x} \, dx \]
Verification is Not applicable to the result.
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fricas [C] time = 0.80, size = 320, normalized size = 1.16 \[ \frac {2 i \, {\rm Ei}\left (\frac {i \, b}{\sqrt {d x + c}}\right ) e^{\left (i \, a\right )} - 2 i \, {\rm Ei}\left (-\frac {i \, b}{\sqrt {d x + c}}\right ) e^{\left (-i \, a\right )} - i \, {\rm Ei}\left (-\frac {2 \, \sqrt {\frac {b^{2} f}{d e - c f}} {\left (d x + c\right )} - 2 i \, \sqrt {d x + c} b}{2 \, {\left (d x + c\right )}}\right ) e^{\left (i \, a + \sqrt {\frac {b^{2} f}{d e - c f}}\right )} - i \, {\rm Ei}\left (\frac {2 \, \sqrt {\frac {b^{2} f}{d e - c f}} {\left (d x + c\right )} + 2 i \, \sqrt {d x + c} b}{2 \, {\left (d x + c\right )}}\right ) e^{\left (i \, a - \sqrt {\frac {b^{2} f}{d e - c f}}\right )} + i \, {\rm Ei}\left (-\frac {2 \, \sqrt {\frac {b^{2} f}{d e - c f}} {\left (d x + c\right )} + 2 i \, \sqrt {d x + c} b}{2 \, {\left (d x + c\right )}}\right ) e^{\left (-i \, a + \sqrt {\frac {b^{2} f}{d e - c f}}\right )} + i \, {\rm Ei}\left (\frac {2 \, \sqrt {\frac {b^{2} f}{d e - c f}} {\left (d x + c\right )} - 2 i \, \sqrt {d x + c} b}{2 \, {\left (d x + c\right )}}\right ) e^{\left (-i \, a - \sqrt {\frac {b^{2} f}{d e - c f}}\right )}}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (a + \frac {b}{\sqrt {d x + c}}\right )}{f x + e}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 438, normalized size = 1.59 \[ -2 b^{2} \left (\frac {\Si \left (\frac {b}{\sqrt {d x +c}}\right ) \cos \relax (a )+\Ci \left (\frac {b}{\sqrt {d x +c}}\right ) \sin \relax (a )}{b^{2} f}-\frac {\Si \left (\frac {b}{\sqrt {d x +c}}+a -\frac {a c f -a d e +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{c f -d e}\right ) \cos \left (\frac {a c f -a d e +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{c f -d e}\right )+\Ci \left (\frac {b}{\sqrt {d x +c}}+a -\frac {a c f -a d e +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{c f -d e}\right ) \sin \left (\frac {a c f -a d e +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{c f -d e}\right )}{2 b^{2} f}-\frac {\Si \left (\frac {b}{\sqrt {d x +c}}+a +\frac {-a c f +a d e +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{c f -d e}\right ) \cos \left (\frac {-a c f +a d e +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{c f -d e}\right )-\Ci \left (\frac {b}{\sqrt {d x +c}}+a +\frac {-a c f +a d e +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{c f -d e}\right ) \sin \left (\frac {-a c f +a d e +\sqrt {b^{2} c \,f^{2}-b^{2} d e f}}{c f -d e}\right )}{2 b^{2} f}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (a + \frac {b}{\sqrt {d x + c}}\right )}{f x + e}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sin \left (a+\frac {b}{\sqrt {c+d\,x}}\right )}{e+f\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin {\left (a + \frac {b}{\sqrt {c + d x}} \right )}}{e + f x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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